Question

A) For AQMNO use the Triangle Proportionality Theorem to solve for x. OMN

Answer 1

• x = 12 units

• perimeter = 113 units

Given for big triangle:

• MN = 36

• MO = 3x + 9 + 15 = 3x + 24

Given for small triangle:

• side 1 : 36 - 9 = 27

• side 2 : 3x + 9

setting up proportionality:

`\hookrightarrow \sf (36)/(3x+24) = (27)/(3x+9)`

`\hookrightarrow \sf 36(3x+9) = 27(3x+24)`

`\hookrightarrow \sf 108x+324 = 81x+648`

`\hookrightarrow \sf 108x-81x = 648-324`

`\hookrightarrow \sf 27x = 324`

`\hookrightarrow \sf x = 12`

Perimeter of the triangle:

36 + 17 + 15 + 3x + 9

36 + 17 + 15 + 3(12) + 9

113 units

Answer 2

x = 12

perimeter = 113 units

Explanation:

Triangle Proportionality Theorem states that if a line parallel to one side of the triangle intersects the other two sides, then it divides the sides into proportional corresponding segments.

`\implies 3x+9 : 15 = (36-9) : 9`

`\implies 3x+9 : 15 = 27 : 9`

`\implies (3x+9)/(15)= (27)/(9)`

`\implies 3x+9 = 45`

`\implies 3x = 36`

`\implies x = 12`

Perimeter =
`3x+9 + 15 + 17 + 36`

Substituting
`x = 12`:

⇒ perimeter = 3(12) + 9 + 15 + 17 + 36

= 113